Monodisperse hard spheres

In the two Newtonian regions, no length or time scales are left in the scaling. Hence, the corresponding relative viscosities 

Ab initio calculations for nondilute systems become very complicated. Einstein derived the linear term in the concentration law for the viscosity in 1906 and 1911. The quadratic term, which is the first interaction term, was published in 1977 by Batchelcn  

The Peclet number of eq. 10.4.6 is based on the difiusivity of an isolated sphere (i.e., Stokes law for the hydrodynamic effect). This is obviously incorrect in a concentrated suspension, where the presence of other particles can have an enormous effect on the mobility. The viscous resistance a particle encounters will be the sum of all hydrodynamic interactions with all neighboring particles. This resistance is much higher than that given by Stokes law and should be comparable with the global viscosity of the suspension. 

Reduced viscosity versus reduced shear stress for suspensions of polystyrene spheres, = 0.50, solid line, in water, open circles, in benzyl alcohol solid circles, m-cresol. Replotted from Krieger (1972). 

On this basis Krieger (1972) has suggested substituting suspension viscosity for the medium viscosity in eq. 10.4.6. The resulting correction of Pe is really a reduced shear stress v, as can be seen in eq. 10.5.3  

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Recent experimental studies (1-3), on systems of sterically stabilized colloidal particles that are dispersed in polymer solutions, have highlighted the role played by the free polymer molecules. These experiments are particularly relevant because the systems chosen are model dispersions in which the particles can be well approximated as monodisperse hard spheres. This simplifies the interpretation of the data and leads to a better understanding of the intcrparticle forces. DeHek and Vrij (1, 2) have added polystyrene molecules to sterically stabilized silica particles dispersed in cyclohexane and observed the separation of the mixtures into two phases—a silica-rich phase and a polystyrene-rich phase—when the concentration of the free polymer exceeds a certain limiting value. These experimental results indicate that the limiting polymer concentration decreases with increasing molecular weight of... 

Dimensional analysis implies that for a given value of , all monodisperse hard-sphere suspensions ought to show an onset of shear thickening at a universal value of the Peclet number Pe, or reduced stress Or. Thus, the critical shear rate Yc foi shear thickening ought... 

In the presence of size polydispersity, there is an additional incoherent contribution to C(, t) decaying through the self-diffusion coefficient Ds((/)) [42,43,91]. The latter can also be measured for monodisperse hard sphere suspensions at finite concentrations at qR corresponding to the first minimum of S( ), i.e., when the interactions can be ignored [91]. These three different diffusion coefficients exhibit distinctly different dependence on q and 0. From these three transport quantities, Z>cou( ) is absent in monodisperse homopolymers, whereas Ds can hardly be measured in polydisperse homopolymers due to the vanishingly small contrast. 

As can be seen from Figure 10.24, v decreases with the increase in (p and ultimately approaches zero when (p exceeds a critical value, maximum packing fraction. The value of tp for monodisperse hard-spheres ranges from 0.64 (for random packing) to 0.74 for hexagonal packing, but exceeds 0.74 for polydisperse systems. For emulsions which are deformable,

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Monodisperse hard-sphere collisions In addition, conservation of momentum implies that... 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

Monodisperse hard-sphere collisions functions about spatial point x yields... 

Concentrated emulsions or high internal phase emulsions (HIPE) are systems in which the volume fraction of the dispersed phase is larger than about 0.74, which is the close-packing volume fraction of monodispersed hard spheres. The dispersed soft entities of a concentrated emulsion are no longer spherical. They deform into polyhedra separated by thin films of continuous phase. The structure is thus analogous to a conventional gas-liquid foam with low liquid content. The structure, properties, stability, and applications of highly concentrated emulsions were recently reviewed by Cameron and... 

In this section, the kinetic aspects of aggregation will especially be discussed. Most of the theory derived is valid for the ideal case of a dilute dispersion of monodisperse hard spheres. Most food dispersions do not comply with these restrictions. Where possible, the effects of deviations from the ideal case will at least be mentioned. Some consequences of aggregation are also discussed. 

To summarize, the dependence of relative viscosity on the volume fraction of suspended particles can be expressed by any of several theoretical or semi-empirical relations. These can be written in terms of the two parameters, [tj] and Thus = t1j.([ii], c )/( ) ). As it will be shown, the generality of this dependence extends beyond the monodispersed hard sphere suspensions. 

The relationships between 17 and ( ) have been derived for suspensions of monodispersed hard spheres in Newtonian liquids. However, most real systems are polydispersed in size, and do not necessarily consist of spherical particles. It has been found that here also Simha s Eq 7.24, Mooney s Eq 7.28, or Krieger-Dougherty s Eq 7.8 are useful, provided that the intrinsic viscosity and the maximum packing volume fraction are defined as functions of particle shape and size polydispersity. For example, by allowing ( ) to vary with composition, it was possible to describe the vs. ( ) variation for bimodal suspensions [Chang and Powell, 1994]. Similarly, after values... 

When the concentration increases, terms higher than linear have to be included in Eq. (16.3). For suspensions of spherical particles a monotonic increase was observed and predicted in the full range of 0 < (p< < max, where < max is the maximum packing volume fraction experimentally, max = 0.62 for monodispersed hard spheres and... 

In the introductory chapter we saw that many systematic depletion studies were performed on mixtures of spherical colloids plus non-adsorbing or free polymers. The reason is obvious spherical colloids are of industrial and fundamental relevance, and can be prepared in a relatively controlled way (rather monodisperse, hard-sphere like), while polymers are ubiquitous, and are efficient depletants. 

The phase behaviour of mixtures of monodisperse hard spheres and polydisperse ideal polymers has been investigated using original FVT [67]. At fixed mean... 

A monodisperse hard sphere is packed in a faee-centered cube with the porosity of 0.295 and orthorhombic system with the porosity of 0.395. The porosities of dry superabsorbent polymers are generally 0.40 to 0.75, and are higher than a hard sphere, for example, a glass bead. However, the permeability of superabsorbent polymers is much lower than glass bends of a similar thiekness. 

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